The two parents transmit the same amount of genetic material to their offspring ....or almost
Nuclear versus Cytoplasmic DNA
All Eukaryotes (everything that is neither a Bacteria nor an Archeabacteria) have nuclear and cytoplasmic DNA. Most of the DNA is nuclear DNA. As its name indicates, nuclear DNA is found in the nucleus. The cytoplasmic DNA is found in the cytoplasm, that is, outside of the nucleus. More specifically, in all eukaryotes, cytoplasmic DNA can be found in the mitochondrion. In plants (which are eukaryotes), you can find cytoplasmic DNA in the mitochondrion and in the plastids (including the chloroplasts (plastids active in photosynthesis)). See here for more information about the origin of the organelles (≈ organ of a cell) who carry DNA.
Nuclear and cytoplasmic DNA doesn't have the same mode of inheritance.
Inheritance of Nuclear Genes
In the diploid (see ploidy) phase (phase during which each individual carries two copies of each gene), an individual carries one copy from his/her mother and one from his/her father.
The above sentence assumes that a diploid phase exists (which is not the case of all species) and that genders (male vs female) exist which is by far not the case of each species. It also assumes that we are talking about autosomes (non-sexual chromosomes, see below for sexual chromosomes). These assumptions hold for humans for example. Note that the diploid phase is much longer in human than the haploid phase (spermatozoid, ovule). We say that humans are diplontic. Some species are haplontic (longer haploid phase) and some are haplo-diplontic. See life-cycle for more info.
In short: Inheritance of nuclear genes is bi-parental, each offspring necessarily carry one copy of a given gene from his/her father and the other copy from his/her mother.
Note: there are some exceptions to this perfect bi-parental transmission (see segregation distorter and intra-genomic conflict). Also, have a look at the short section on sexual chromosomes below.
Inheritance of Cytoplasmic Genes
There are many exceptions to what I am about to write...
Globally speaking, offspring inherit cytoplasmic DNA for their mothers only. That means that your mitochondrion is those of your mother only. Therefore, mothers transmit slightly more genes than fathers because of the concern of cytoplasmic genes.
To complete the picture...
Inheritance of Sexual Chromosomes (Nuclear DNA)
Not all sexual species have sex chromosomes but some do. In these species, it is not rare that one of the two sexual chromosomes is shorter than the other (In mammals, the $Y$ chromosome is shorter than the $X$ chromosome). In such a case, the male transmits slightly less DNA to their male offspring than to their female offspring. Also, in such case, fathers transmit less DNA to their male offspring than do mothers but males transmit as much DNA to their female offspring than do mothers.
The evolution of sex chromosomes is a very interesting subject as well and you will find tons of information on this website or on different sites.
Nutrient in eggs
Not only genes are transmitted. Parents also transmit proteins and nutrients in the egg. Females transmit more nutrients to their offspring than males.
Not only genes are transmitted. Parents also transmit modifications on the DNA that are not modifications in the DNA sequence per se. These modifications are called "epigenetic modifications". Not all species can produce epigenetic modifications. In species who can transmit epigenetic modifications, both males and females can produce them. It is possible that in some species mother have a higher variance in the modifications that they transmit than males (or vice-versa) but I am not aware of such things.
Environment and Cultural transmission
Not only genes are transmitted. Parents also offer an environment in which the offspring grow. Some species typically feed or protect their offsprings. Depending on the species males or females may have more influence on the environment experienced by the offsprings. Finally, in some species (human, chimps and orca for example) a set of traditions (culture) are transmitted to the offsprings from the parents. And depending on the species the father or the mother may have more influence in the culture transmitted to the offspring.
Length of chromosomes may differ between two randomly sampled individuals
Not all variants between two chromosomes in a population is caused by substitutions of one nucleotide by another. Some variation is due to deletions of nucleotides or insertions of nucleotides. You can even a very long piece of DNA who get inserted, duplicated or deleted. Therefore between two given individuals, the exact amount of DNA that is transmitted may also differ only because they happen to have different variants. However, this is not a systematic difference in the quantity of DNA that is transmitted by one sex or another unlike the other cases presented above.